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Revenue Ranking of Discriminatory and Uniform Auctions
with an Unknown Number of Bidders
Aleksandar Saša Pekeč, Ilia Tsetlin,
To cite this article:
Aleksandar Saša Pekeč, Ilia Tsetlin, (2008) Revenue Ranking of Discriminatory and Uniform Auctions with an Unknown Number
of Bidders. Management Science 54(9):1610-1623. http://dx.doi.org/10.1287/mnsc.1080.0882
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Vol. 54, No. 9, September 2008, pp. 1610–1623
issn 0025-1909 eissn 1526-5501 08 5409 1610
®
doi 10.1287/mnsc.1080.0882
© 2008 INFORMS
Revenue Ranking of Discriminatory and Uniform
Auctions with an Unknown Number of Bidders
Aleksandar Saša Pekeč
The Fuqua School of Business, Duke University, Durham, North Carolina 27708,
[email protected]
Ilia Tsetlin
INSEAD, 138676 Singapore, [email protected]
A
n important managerial question is the choice of the pricing rule. We study whether this choice depends
on the uncertainty about the number of participating bidders by comparing expected revenues under
discriminatory and uniform pricing within an auction model with affiliated values, stochastic number of bidders,
and linear bidding strategies. We show that if uncertainty about the number of bidders is substantial, then the
discriminatory pricing generates higher expected revenues than the uniform pricing. In particular, the first-price
auction might generate higher revenues than the second-price auction. Therefore, uncertainty about the number
of bidders is an important factor to consider when choosing the pricing rule. We also study whether eliminating
this uncertainty, i.e., revealing the number of bidders, is in the seller’s interests, and discuss the existence of an
increasing symmetric equilibrium.
Key words: discriminatory pricing; uniform pricing; auctions; demand uncertainty; stochastic number of
bidders
History: Accepted by David E. Bell, decision analysis; received February 22, 2007. This paper was with the
authors 2 months for 2 revisions. Published online in Articles in Advance July 21, 2008.
1.
Introduction
time ago in the context of auctions for oil leases
(e.g., Capen et al. 1971, Engelbrecht-Wiggans et al.
1986). This uncertainty is natural in sealed-bid auctions; for example, it is an important consideration in
design of spectrum auctions (Klemperer 2004, §5.6.2).1
In auctions that are conducted electronically, bidders are also unlikely to know the number of their
competitors. Arora et al. (2007) highlight the prevalence of uncertainty about the number of competitors in electronic marketplaces. Electronic auctions are
widespread in business transactions: from direct sales
to customers (such as millions of auctions conducted
daily at eBay.com) to becoming a standard transaction
format in many supply chains and distribution channels (e.g., Elmaghraby 2007 surveys current industry practice).2 Given the prominence of auctions in
business and, consequently, in management science
research (e.g, see Geoffrion and Krishnan 2003, for
the overview of a special issue of Management Science
The choice of the pricing rule is an important and
complex managerial decision. The focus of this paper
is on the effect of demand uncertainty, within an
auction model, on expected revenues under uniform
pricing (everyone pays the same price) and discriminatory pricing (price varies by customer). Specifically,
we study how uncertainty about the number of bidders impacts the choice of the pricing rule.
Auctions are not only widely used to make allocation and pricing decisions in competitive environments where all involved parties act strategically in
their own best interests, but auction theory also provides a framework for analytical study of pricing and
allocation. Although previous research has identified
a number of factors that make a particular auction format (and thus the pricing rule) more attractive to the
seller, most of this work has assumed that the number of bidders participating in an auction is known at
the moment of bid submission. This is not realistic,
because uncertainty about the number of bidders is
common: both the bid-taker at the time of making the
decision on the pricing rule, and bidders at the time
of submitting the bids, might not know how many
bidders are participating.
The uncertainty about the number of bidders and
its impact on the auction outcome was noted some
1
Levin and Ozdenoren (2004, p. 230) point that “In the last spectrum auction in Great Britain that raised about $35 billion the issue
of the number of competing companies was so central that the auction form was designed mainly with that in mind.”
2
Most frequently auctions are considered in the setting with a single seller and multiple buyers. In the case of procurement auctions
there is a single buyer and multiple sellers. However, from a gametheoretic point of view, these two settings are equivalent.
1610
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Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
on e-business, and Anandalingam et al. 2005, for the
overview of a special issue of Management Science on
electronic market design), it is important to understand the effect of uncertainty about the number of
bidders on the decisions of auction designers and auction participants.
In this paper, we show that demand uncertainty,
modeled here as the uncertainty about the number of
bidders, is an important factor that cannot be overlooked. Even for the simplest auction formats, standard revenue rankings of auction pricing rules do not
necessarily hold when there is uncertainty about the
number of bidders.
We analyze single-round sealed-bid auctions in
which every bidder demands one object. If k identical
objects are allocated through the auction procedure,
the top k bidders get one object each. We consider two
pricing rules: discriminatory pricing (discriminatory
auction), in which every winning bidder pays his bid,
and uniform pricing (uniform auction), in which all
winning bidders pay the same price.3 Discriminatory
and (variants of) uniform pricing are the most prevalent pricing rules in practice. For k = 1, i.e., one object,
these two pricing rules correspond to the first-price
and the second-price auctions.
A theoretical auction model specifies the information that bidders have and the structure of bidders’
valuations: each bidder observes a signal about the
object value and his valuation for an object depends, in
general, on his signal and on signals of the other bidders. The auction literature distinguishes between two
polar cases of bidders’ valuations—common value
and private value settings: in the former the value of
the object is the same for all bidders, and in the latter
it depends only upon a bidder’s signal. Thus, in the
common-value case, all bidders observe different estimates of the (unknown) object value; in the privatevalue case bidders know their own values, but not
the values of others. Bidders’ signals can be independent or affiliated, where affiliation implies that higher
signals of any of the bidders make higher signals of
other bidders more likely.
There is a large auction theory literature that deals
with revenue comparisons of different auction formats in different theoretical settings (e.g., Rothkopf
and Harstad 1994 provide a review). However, all of
that work assumes no uncertainty about the number of
bidders. A classical model of uniform and discriminatory auctions is that of Milgrom (1981). In this model,
bidders are risk neutral, and bidders’ signals are affiliated. Milgrom and Weber (1982) and Jackson and
Kremer (2006) prove a “standard revenue ranking”
result: the uniform auction yields greater expected
3
As standard in auction theory, we assume that the uniform price
is set at the value of the highest losing bid.
1611
revenue than the discriminatory auction in the setting with affiliated values and risk-neutral bidders.
In the special case of valuations being independent
and drawn from the same distribution (e.g., independent private values), the revenue equivalence theorem
holds and the expected revenues from discriminatory and uniform auction are equal (Maskin and Riley
1989). However, discriminatory auctions could yield
higher revenues than uniform auctions if the riskneutrality assumption or the unit-demand assumption is violated. If buyers are risk averse and the seller
is risk neutral, discriminatory pricing is better for the
seller than uniform pricing (Holt 1980, Maskin and
Riley 1984). Back and Zender (1993) show that if bidders demand more than one object, there exist collusive strategies in the uniform price auction, and argue
that discriminatory auctions might therefore be more
profitable for the seller.
Uncertainty about the number of bidders can be
modeled as endogenous or exogenous. In the endogenous case, each bidder decides whether to participate or not. That decision would depend on, e.g.,
entry fee, cost of preparing the bid and of acquiring necessary information, and reservation price
(Engelbrecht-Wiggans 1987, Harstad 1990, Levin and
Smith 1994, Samuelson 1985, Hendricks et al. 2003,
Li and Zheng 2006, and McAfee et al. 2002 also consider empirical implications). In the exogenous case,
the number of bidders is drawn from a distribution
specifying the probability that a particular number
of bidders participate in the auction. To isolate the
effect of the uncertainty about the number of bidders
on revenue rankings of standard auction formats, our
model eliminates other parameters such as entry fees,
reserve prices, bid preparation costs, etc. Thus, we
model uncertainty about the number of bidders as
exogenous.
A model with an unknown (exogenous) number
of bidders was introduced by Matthews (1987) and
McAfee and McMillan (1987). Both of these papers
focus on the case of independent private values and
risk-averse bidders. Matthews (1987) also considers
the case of affiliated private values, but does not compare auction revenues in first- and second-price auctions when the number of bidders is unknown. Levin
and Ozdenoren (2004) consider independent private
values and ambiguity-averse bidders. Harstad et al.
(1990) consider the model with independent private
information and risk-neutral bidders, and characterize equilibrium bidding strategies. In that case, the
revenue equivalence theorem holds and the revenue
in the first- and second-price auctions is the same.
Furthermore, the auction revenue does not depend
on whether the number of bidders is revealed or
concealed.
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1612
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
To study the impact of the uncertainty about the
number of bidders on the revenue ranking of discriminatory and uniform pricing, we combine the classical model of Milgrom (1981) with Matthews’ (1987)
model for an uncertain number of bidders. Our model
is presented in §2. It poses several challenges with
respect to determining equilibrium behavior. Thus,
we conclude §2 by introducing an assumption that
will allow for analytical tractability when comparing
pricing rules in §§3 and 4, and we relegate discussion
about equilibrium bidding strategies to Appendix A.
If uncertainty about the number of bidders is not
large in the sense that all possible numbers of bidders
are sufficiently close to each other, the revenue ranking should correspond to the case where the number
of bidders is known and the choice of the pricing
rule will not depend on such small uncertainty. In
that sense, small uncertainty can be ignored by the
seller. However, if uncertainty about the number of
bidders is large enough, the revenue ranking of discriminatory and uniform pricing gets reversed. Thus,
if uncertainty about the number of bidders is substantial, it is an important factor to consider when
choosing the pricing rule. Section 3 presents a formal
theorem, an illustration, and a discussion.
In some situations, the bid-taker might be able to
resolve uncertainty about the number of bidders by
publicly releasing the exact number of bidders before
bids are submitted (e.g., bidders might be required to
register or place a deposit before the auction starts).
Section 4 studies the effect of resolving uncertainty
about the number of bidders on expected revenues.
For example, Matthews (1987) shows that, in the case
of affiliated private values, the seller benefits from
concealing the number of bidders in the discriminatory auction. We confirm this result, and also show
that, with common values, the seller might prefer to
reveal the number of bidders. More generally, our
main result in §4 is that by resolving uncertainty
about the number of bidders prior to bid submission, the bid-taker benefits more in the uniform auction than in the discriminatory auction. Thus, all other
things being equal, the bid-taker has a stronger incentive to reduce this uncertainty under uniform pricing
than under discriminatory pricing.
Section 5 concludes. Appendix A discusses the issue
of equilibrium existence. All proofs are in Appendix B.
2.
Model with Linear Bidding
Functions
We first generalize Milgrom’s (1981) and Pesendorfer
and Swinkels’ (1997) common-value auction model
with unit demands, by allowing the exact number of
bidders to be unknown at the moment of bid submission and by allowing for a mixture of common
and private valuations. Then we introduce Assumption 1, which ensures linearity of bidding functions,
and state the linear bidding strategies that are used
in the analysis in §§3 and 4.
We consider uniform and discriminatory sealedbid auctions: k homogeneous objects are sold to
the k highest bidders. Ties, if any, are broken randomly. In a discriminatory auction, the winning bidders pay their bids, whereas in a uniform auction,
all winning bidders pay a uniform price equal to the
(k + 1)st-highest bid. When k = 1, discriminatory and
uniform auctions reduce to first-price and secondprice auctions, respectively.
The uncertainty about the number of bidders is
modeled following Matthews (1987). Bidders are
drawn from a pool of N potential bidders in accordance with an exogenous stochastic process, =
n1 1 nM M , specifying that ni bidders are
present with probability i > 0. As is standard with
information about an auction setting, is assumed
to be common knowledge. The case with no demand
uncertainty (i.e., n bidders for sure) corresponds to
= n 1. We also assume that the number of
objects is always less than the number of bidders: 1 ≤
k < minn1 nM .
All potential bidders are risk neutral. There is one
state variable V with commonly known probability density function (p.d.f.) gv with support v v.
Bidder j independently observes an estimate Xj , a
random scalar distributed with p.d.f. f x v and
cumulative distribution function (c.d.f.) F x v, conditional on V = v. The support of the marginal distribution of Xj is x x, − < x < x ≤ +.
As in Milgrom (1981), f x v satisfies the monotone likelihood-ratio property (MLRP)
f x v
f x v
≥
f x v f x v ∀x > x v > v (1)
so that the Xj s and V are affiliated. For example, if Xj
is normally (or uniformly) distributed with mean v,
then (1) is satisfied. Furthermore, a truncated normal
distribution would also satisfy MLRP.
The value of one object for bidder j is given by
Vj = uV Xj , where the function u· is continuous
and increasing in both variables. That description
incorporates private values and common values as
special cases: If uV X = V , then this is the commonvalue model (used by Milgrom 1981 and Pesendorfer
and Swinkels 1997, dating back to Rothkopf 1969 and
Wilson 1969), where the value of the object is the same
for all bidders and unknown at the moment of bidding. If uV X = X, then this is the affiliated privatevalues model, where each bidder knows his valuation
for the object. Finally, if the distribution of V is degenerate, i.e., it assigns probability one to a single point,
then this is the independent private-values model.
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Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
Now we derive the symmetric equilibrium bidding
functions in the uniform and discriminatory auctions.
It will be useful to take the point of view of one of the
bidders, say bidder 1 with signal X1 = x, and to consider the order statistics associated with the signals of
all other bidders. We denote the mth-highest signal of
bidders 2 3 ni (i.e., all active bidders except bidk
given x by
der 1) by Ynmi −1 , the conditional p.d.f. of Yn−1
fn−1k y x =
v
·
v
n−1!
k−1!n−k−1!
f y vF n−k−1 y v1−F y vk−1 f x vgv dv
v
f x vgv dv
v
(2)
and the corresponding c.d.f. by Fn−1 k y x.
We start by stating the symmetric equilibrium bidding function in a uniform auction. Define
k
= y
vnk xy = EV1 X1 = xYn−1
v
=
v
k−1
uvxf y vF n−k−1 y v 1−F y v
f x vgv dv
k−1
v
n−k−1
f y tF
y t 1−F y t
f x tgt dt
v
(3)
and
M
vxy =
i=1 i ni vni k xyfni −1k y x
M
i=1 i ni fni −1k y x
(4)
where fn−1 k y x is defined by (2). If the number
of bidders is known, then the symmetric equilibrium
bidding function in a uniform auction is given by
vnk x x (Milgrom 1981).
Theorem 2.1. If there exists a symmetric equilibrium
b u : → in increasing strategies for a uniform auction,
then it is
(5)
b u x = vx x
where vx x is defined by (4). If b u is increasing, then it is
the unique symmetric equilibrium in increasing strategies.
Before describing the symmetric equilibrium bidding function for the discriminatory auction, denote
M
i=1 i ni fni −1 k y x
Ay x = M
(6)
i=1 i ni Fni −1 k y x
Theorem 2.2. If there exists a symmetric equilibrium
b d " → in increasing differentiable strategies for a discriminatory auction, then it is
x
t
vt tAt te x As s ds dt
(7)
b d x =
x
where vt t is defined by (4) and At t is defined by (6).
1613
To make revenue comparisons of discriminatory
and uniform pricing analytically tractable, our analysis in §§3 and 4 is limited to frameworks that satisfy
Assumption 1.
Assumption 1.
(i) There exists %, 0 ≤ % ≤ 1, such that uv x = %v +
1 − %x.
(ii) f xv is location invariant, i.e., f xv = hx − v,
where h· is a p.d.f. (and the corresponding c.d.f. is
denoted by H).
(ii ) The support of ht is t1 t2 , and mint∈t1 t2 ht =
' > 0.
(iii) The p.d.f. gv is uniformly distributed on −T T with T → .
Assumption 1(i) allows for a mixture of privatevalue and common-value components: % = 0 corresponds to the affiliated private-values model, and
% = 1 corresponds to the common-value model.
Assumption 1(ii) is equivalent to assuming that
X = V + Z, where Z is some noise term, independent
of V . Many important signal distributions satisfy (ii);
(ii ) is a technical assumption to simplify the proof
of Theorem 3.1. Assumption 1(iii) corresponds to a
diffuse prior assumption in Bayesian statistics: public information about V , specified by gv, is much
less precise than private information, specified by
f x v. Its use in auction theory traces back to Wilson (1969, §4), and was further discussed in Rothkopf
(1980a, b). The setting with a diffuse prior is also
extensively used in the experimental literature (Kagel
et al. 1987, Kagel and Levin 2002, Parlour et al. 2007).4
Note that the posterior (i.e., conditional on signal
v x) density of V is given by gv x = f x vgv/
f x vgv dv. Thus, the diffuse prior assumpv
v
tion (iii) implies that gv x = f x v/ v f x v dv.
Using the locationinvariance assumption (ii), we get
v
gv x =hx − v/ v hx − v dv = hx − v. Therefore,
(ii) and (iii) together imply that the p.d.f. of the posterior distribution, conditional on signal x, is also location invariant.
4
Parlour and Rajan (2005) consider the situation where random
variable V has finite support, rather than being diffuse over the real
line. This situation has the advantage of making the model more
realistic (e.g., V is bounded above and below), but comes at the
cost of having to resort to numerical methods to calculate bidding
functions near the boundary of the signal support. In the interior
of the signal support, on the other hand, the bidding functions
correspond to the analytically solvable case studied here.
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
1614
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
Using the notation
+nk =
n − 1!
k − 1!n − k − 1!
+
·
tH n−k−1 t1 − H tk−1 h2 t dt
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−
-nk =
Propositions 2.3 and 2.4 provide bidding functions
that will be used in §§3 and 4, and are special cases
of Theorems 2.1 and 2.2.
n − 1!
k − 1!n − k − 1!
+
·
H n−k−1 t1 − H tk−1 h2 t dt
(8)
−
we can describe the symmetric equilibrium bidding
functions. In the standard case where the number of
bidders n is fixed, the bidding function in the uniform
auction is
+
u
bnk
x = x − % nk (9)
-nk
The term −%+nk /-nk can be understood as a winner’s curse correction. As Proposition 2.3 shows, if the
number of bidders is unknown, the bidding function
is a weighted average of the bidding functions with
known numbers of bidders, where the weights correspond to the probabilities of ni bidders conditional on
the assumption that a given bidder is tied with the
kth-highest rival.
Proposition 2.3. Under Assumption 1, the unique
increasing symmetric equilibrium in a uniform auction is
M
i ni -ni k
M
bnui k x
i=1
j=1 j nj -nj k
M
i=1 i ni +ni k
= x − % M
i=1 i ni -ni k
b u x =
(10)
In the discriminatory auction with known number
of bidders, the bidding function is
d
bnk
x = x − %
+nk
k
−
-nk n-nk
(11)
The term −k/n-nk reflects underbidding in the discriminatory auction, relative to bidding function in
the uniform auction. Equation (11) agrees with the
equilibria described by Winkler and Brooks (1980,
p. 610; they have k = 1, n = 2, and a normal signal distribution h·); and by Klemperer (1999, p. 260;
he has k = 1 and uniform h·); see also Klemperer
(2004, p. 57). The next proposition describes the bidding function for a discriminatory auction.
Proposition 2.4. If there exists a symmetric equilibrium b d : → in increasing differentiable strategies for
a discriminatory auction under Assumption 1, then it is
M
i ni -ni k
M
bndi k x
n
i=1
j
j
n
k
j=1
j
M
k
i=1 i ni +ni k
= x − % M
− M
i=1 i ni -ni k
i=1 i ni -ni k
b d x =
(12)
3.
Revenue Comparisons
As mentioned in the introduction, the auction theory
literature has identified many important factors influencing the choice of the pricing rule. In this paper,
we identify another one: uncertainty about the number of bidders. Intuitively, the uncertainty that really
matters here is that both relatively large and relatively small numbers of bidders are possible. We show
that sufficiently changing the number of bidders in
just one of the states of the stochastic process (i.e., increasing demand without changing the probability of that state, no matter how small that probability is), results in reversal of the standard revenue
ranking. Our main result is the following. Consider
any distribution over the number of bidders, specified by = n1 1 nM M . Keep all parameters (i.e., 1 M , n1 nM−1 ) fixed except for nM .
Then, for large enough nM , the discriminatory auction
yields greater revenues than the uniform auction. As
mentioned in the introduction, the standard revenue
ranking result (when there is no uncertainty about the
number of bidders) implies that the uniform auction
yields greater expected revenues. Thus, for any probability distribution over the number of bidders, the
standard revenue ranking will be reversed provided
the number of bidders is large enough in one of the
states (i.e., if both relatively large and relatively small
numbers of bidders are possible).
Theorem 3.1. For any auction setting satisfying
Assumption 1, any number of objects k, and any
1 M , n1 nM−1 , there exists n∗ such that, for
any nM ≥ n∗ , the discriminatory auction for k objects
yields greater expected revenue than the uniform k + 1st
price auction.
3.1.
Illustration with Uniform Distribution
of Signals
To understand the intuition underlying Theorem 3.1,
consider an example with a uniform distribution of
signals: conditional on V = v, each bidder’s signal is
drawn independently from a uniform distribution on
v − 12 v + 12 . This important example corresponds
to Assumption 1 with the signal distribution h·
being uniform on − 12 12 , and has been extensively
used in the experimental literature to study firstprice, second-price, and English auctions in the two
polar cases of pure private (% = 0) and pure common values (% = 1); see Kagel et al. (1987) and Kagel
and Levin (2002). It is the best known example in
which a tractable solution can be obtained. Klemperer
(2004, pp. 55–57) presents the equilibria and revenue
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Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
comparisons of first-price, second-price, and English
auctions for the pure common-value case in which
% = 1. With uniform signals, +nk and -nk , defined
in (8), are +nk = 12 − k/n and -nk = 1.
Another advantage of this example is that, for the
pure common-value case, the bidding function in the
discriminatory auction does not depend on the number of bidders: it is b d x = x − 12 . Thus, even though
Proposition 2.4 states only the necessary condition for
an increasing symmetric equilibrium in a discriminatory auction with an unknown number of bidders,
the proof that it is an equilibrium is the same as for
the case with a known number of bidders. Therefore, Assumption 1, with uniform signals and % = 1,
provides an example of an auction setting where the
increasing symmetric equilibrium, stated in Proposition 2.4, exists.
Consider a common-value auction for one object
(i.e., k = 1 and % = 1), in which either n1 or
n2 bidders participate with equal probability, i.e.,
= n1 12 n2 12 . As Klemperer (2004, pp. 56–57)
shows, when the number of bidders n is known,
the bidding functions in the first-price and secondd
u
price auctions are bn1
x = x − 12 and bn1
x = x −
1
+
1/n.
When
the
number
of
bidders
is
unknown,
2
the bidding function in the first-price auction is the
same, b d x = x − 12 . (This follows from Proposition 2.4,
because the bidding function with an unknown number of bidders is a linear combination of the bidding
functions with known numbers of bidders.) In the
case of the second-price auction, the bidding func nu1 1 x + 2 n2 /nb
nu2 1 x =
tion (10) is b u x = 1 n1 /nb
1
where n = 1 n1 + 2 n2 , the expected numx − 2 +1/n,
ber of bidders.
Consider the case where n1 = 2 and n2 is very
large. Then n is very large as well, and therefore b u x ≈ x − 12 , i.e., bids in the first-price and
the second-price auctions almost coincide. Now, if
the realized number of bidders is n2 , the signal of the
price setter, conditional on V = v, will be very close
to v + 12 , and thus the auction price will be very close
to V in both first-price and second-price auctions.
However, if the realized number of bidders is n1 = 2,
the expected value of the price setter’s signal, conditional on V = v, is v + 16 in the case of the first-price
auction and v − 16 in the case of the second-price auction. Therefore, if the number of bidders is n1 = 2,
the expected selling price in the first-price auction is
higher by 13 than in the second-price auction.
The result above is due to the fact that although
the probability of n1 = 2 bidders is one-half from the
seller’s perspective, bidders bid as if this probability is much lower: b u x ≈ x − 12 ≈ bnu2 1 x. Conditional
on being active, a bidder updates the probabilities
of n1 =2 and n2 bidders and finds that the greater
1615
number of bidders (i.e., n2 ) is much more likely.5
Indeed, in the case of ni active bidders, the probability that a given bidder participates in the auction is
given by ni /N . Thus, by Bayes Theorem, the probability of ni bidders from
the perspective of an active
bidder is i ni /N / M
i=1 i ni /N = i ni /n.
Theorem 3.1 is due to exactly this effect. As nM
becomes large, active bidders put disproportionately
large weight on nM bidders being active, irrespective of the underlying probabilities 1 M . That
does not hurt the seller in the case of nM bidders
being active, because (for the common-value case)
the selling price is very close to V in both uniform and discriminatory auctions. However, if the
number of active bidders is not large, the difference
between price-setting signals (for first-price versus
second-price auctions, we are interested in the difference between the highest and the second-highest
signals) is significant.
Note that an extreme uncertainty setting in the discussion above is convenient for understanding intuition, and for proving Theorem 3.1 for any auction
setting. However, for any given signal distribution, it
is easy to construct examples where the uncertainty
about the number of bidders does not take such an
extreme form. For the uniform distribution of signals
discussed above, the difference between the expected
revenues of uniform (Ru ) and discriminatory (Rd ) auctions is6
M
i
1
1 + 1/k u
d
2
ER − R = k M
−
2
n +1
i=1 i
i=1 i ni
If the number of bidders is either n1 or n2 with
equal probabilities, then the discriminatory auction
of one object (i.e., k = 1) generates greater revenues than the corresponding
uniform auction if
and only if n2 > n1 + 1 + 4n1 + 5 (e.g., n1 = 10 and
n2 = 18, or n1 = 100 and n2 = 122). If the number
of bidders is uniformly distributed from n1 to nM
(i.e., 1 = 2 = · · · = M = 1/nM − n1 + 1), then the
discriminatory auction of one object generates greater
revenues than the corresponding uniform auction for,
e.g., n1 = 2 and nM = 10, n1 = 10 and nM = 24, n1 = 100
and nM = 137.
The examples above show that revenue-ranking
reversal described in Theorem 3.1 occurs even with
moderate levels of the uncertainty about the number
of bidders. However, if the level of uncertainty about
the number of bidders were low enough, the revenue ranking of uniform and discriminatory auctions
would coincide with the standard revenue ranking.
5
The underlying intuition is also discussed in Matthews (1987) and
Harstad et al. (1990).
6
That expression follows from (B16), which appears in the proof of
Theorem 3.1 in Appendix B.
1616
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4.
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
Revealing the Number of Bidders
Depending on the auction format, neither bidders
nor the bid-taker might know the exact number of
bidders prior to bid submission. In some situations,
the bid-taker might first register/approve bidders
(e.g., by requesting that bidders deposit a considerable amount of cash, as is commonly done in government auctions such as FCC spectrum auctions),
publicly release the number of participating bidders,
and only then solicit bid submissions. Of course, the
bid-taker might choose not to release this information.
In this section, we investigate the impact of resolving uncertainty about the number of bidders prior
to bid submission by comparing auction revenues in
auctions with known and unknown numbers of bidders. The auction theory literature provides answers
in some special cases: if bidders’ signals are independent and bidders are risk neutral, the revenue equivalence theorem holds, and therefore the expected
revenue does not depend upon whether the actual
numbers of bidders and objects are revealed or concealed (Harstad et al. 1990, Klemperer 1999). If bidders are risk averse and have private valuations, then,
as shown by Matthews (1987), revealing the actual
number of bidders decreases the seller’s revenue
in discriminatory auctions and does not change the
expected revenue in uniform price auctions. Harstad
et al. (2008) show that in the uniform auction with
pure common values, for a large enough number of
bidders, the seller benefits by revealing the number of
bidders.
Our analysis in this section allows for affiliated and
nonprivate valuations, and we show that recommendations for the special cases mentioned above cannot be generalized: even in the setting limited by
Assumption 1, the effect of revealing the information
about the number of bidders on discriminatory auction revenues can be positive or negative.
Example 4.1. Consider a discriminatory auction for
one object, in which either two or three bidders
participate with equal probabilities, i.e., let =
2 0
5 3 0
5. Let ht = 1 + at, −1/2 ≤ t ≤ 1/2,
−2 ≤ a ≤ 2.
Figure 1 plots the difference between the expected
revenues in the discriminatory auction with unknown
and known numbers of bidders (ERd − ERdK ) for
different values of %. As shown in the figure, the
seller benefits from concealing the number of bidders
if % < 0
5 for any a. However, in the pure commonvalue case (i.e., % = 1), the seller benefits from concealing the number of bidders when a > 0, and from
revealing the number of bidders when a < 0.
Even though the effect of resolving uncertainty
about the number of bidders is ambiguous, resolving this uncertainty under uniform pricing is always
more beneficial than resolving it under discriminatory
Figure 1
The Difference in Expected Auction Revenues with Unknown
and Known Number of Bidders in a Discriminatory Auction,
ERd − ERKd as a Function of Parameter a of the Signal
Distribution, for Different Values of the Common-Value
Component 0.03
δ=0
δ = 0.25
0.02
δ = 0.50
δ = 0.75
δ = 1.00
0.01
0.00
a
–0.01
–2.0
–1.5
–1.0
–0.5
0
0.5
1.0
1.5
2.0
Note. Model parameters are specified in Example 4.1.
pricing. Theorem 4.2 compares the effect of revealing
the number of bidders in uniform and discriminatory
auctions.
Theorem 4.2. Consider an auction setting that satisfies
Assumption 1. By making the number of bidders known
prior to bid submission, the seller benefits more in the uniform auction than in the discriminatory auction.
The following corollary is a direct consequence of
the fact that for private values (i.e., for % = 0), the
expected revenue in uniform auctions for known and
unknown numbers of bidders is the same (because
b u x = bnui k x = x), and it corresponds to a result from
Matthews (1987, Theorem 4).
Corollary 4.3. For any auction setting that satisfies
Assumption 1 with % = 0, the seller benefits by concealing
the number of bidders in the discriminatory auction.
The intuition behind Theorem 4.2 and Corollary 4.3
is the following. Revealing the number of bidders
helps the seller if the bidding function with a known
number of bidders is decreasing in n. From (11) with
d
% = 0, bnk
x is increasing in n, and thus, if values are
private, the seller benefits by concealing the number
d
u
of bidders.7 Also, because bnk
x = bnk
x − k/n-nk ,
by revealing the number of bidders, the seller benefits
more in the case of the uniform auction.
As Corollary 4.3 shows, in the discriminatory auction the seller benefits by concealing the number
of bidders if bidders’ valuations are private (and
Assumption 1 is satisfied). With nonprivate values,
the bidding function in the discriminatory auction
with known number of bidders might be increasing
7
Harstad et al. (2008) prove that asymptotically (i.e., for large k
and n) n-nk and +nk /-nk are increasing in n.
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Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
or decreasing in n, and thus Theorem 4.2 suggests
that the revenue ranking is ambiguous. As shown in
Example 4.1, the revenue ranking indeed depends on
the signal distribution.
In summary, a decision to mitigate or induce uncertainty about the number of bidders could have impact
on revenues, and the optimal policy depends on the
information and valuation structure of the bidders.
However, our findings indicate that the benefit of mitigating demand uncertainty is higher in the context of
uniform pricing than in the context of discriminatory
pricing.
5.
Conclusions
We analyze the impact of uncertainty about the number of bidders on the expected revenues of uniform
and discriminatory pricing rules in the single-shot
unit-demand auction model that allows for an uncertain number of bidders and for affiliated values. We
show that the revenue ranking gets reversed if uncertainty about the number of bidders is sufficiently
large (Theorem 3.1). In other words, although uniform auctions are known to generate higher expected
revenues than discriminatory auctions when there is
no uncertainty about the number of bidders, the discriminatory auctions yield greater expected revenue
than the uniform auctions if that uncertainty is large.
Intuitively, this result follows from the fact that bidders rationally bid as if the greater number of rivals
is more likely.
Section 4 analyzes the circumstances under which
it is beneficial to mitigate or induce uncertainty by
revealing or concealing the information about the
number of bidders. There is no clear-cut answer, and
it depends on whether the equilibrium bidding function with a known number of bidders is increasing in
the number of bidders. If it is decreasing (as in the
uniform auction), the seller should reveal the number
of bidders. If it is increasing (as in the discriminatory
private-values auction), the seller benefits from concealing the number of bidders.
Our approach required extending a classical auction theory model to account for uncertainty about the
number of bidders. As demonstrated in Appendix A,
an increasing equilibrium in such an extended model
might not exist, and proving equilibria results in
full generality becomes analytically intractable. These
technical issues point to one weakness of our results:
in the case of discriminatory auctions with an unknown number of bidders, we work only with the
candidate for an increasing symmetric equilibrium.
When the symmetric equilibrium fails to exist, there
might be no natural equilibrium to focus on, and
the revenue ordering might not be well defined, due
to, e.g., existence of multiple equilibria. On the other
1617
hand, the setting with linear bidding strategies, specified in Assumption 1, is particularly amenable to
experimental testing, as discussed in §3.1. Studying
whether the subjects bid according to the strategies
specified here, and whether model predictions hold
in the laboratory setting, could lead to interesting
new insights about the preferable auction format and
the predictive power of Bayesian Nash equilibrium
theory.
The impact of uncertainty about the number of bidders on revenue rankings, even in a setting limited
by Assumption 1, implies that this uncertainty is an
important factor to consider when choosing the auction pricing rule, and if that uncertainty is substantial,
discriminatory pricing is preferable for the seller. To
the extent that the auction setting is representative of
more general competitive environments, our results
suggest that, overall, the choice of uniform versus discriminatory pricing is sensitive to the level of demand
uncertainty.
Acknowledgments
The authors are grateful to Claudio Mezzetti, Michael
Rothkopf, Robert Winkler, the associate editor, and the referees for helpful and insightful comments. The second author
gratefully acknowledges the financial support of INSEAD
Alumni Fund.
Appendix A. Existence of an Increasing
Symmetric Equilibrium
Theorems 2.1 and 2.2 provide the unique candidates, but
do not guarantee the existence of an increasing symmetric equilibrium. The necessary and sufficient condition from
Theorem 2.1 implies that in the case of a private-values
model, where uv x = x, the bidding function in a uniform
auction, b u x = x, is increasing, and thus an increasing symmetric equilibrium exists. Example A.1 shows that in the
case of a common-value model, where uv x = v, a symmetric increasing equilibrium might not exist. In the case of
the discriminatory auction, an increasing symmetric equilibrium might not exist for two reasons: (i) bidding function, given by (7), might not be increasing (Example A.1);
and (ii) even when it is increasing, deviations from it might
be profitable (Example A.2). To simplify the exposition, we
provide the examples only for k = 1 and for the cases where
uv x = v or uv x = x. However, similar examples can
be constructed for k > 1 and for a more general form of
valuation function uv x.
Example A.1. Consider an auction for one object (i.e.,
k = 1) with k = 2 1 1/2 n2 1 1/2; that is, there
are either 2 or n2 bidders, each equally likely. Let
gv be uniform on 0 v, v > 1, let f x v be uniform on v − 1/2 v + 1/2, and let uv x = v. Consider
−1/2 < x < 1/2. As shown below, b u x, given by (5), is not
increasing for n2 ≥ 6. Figure A1 shows that b d x, given
by (7), is not increasing for n2 ≥ 7. Therefore, an increasing
symmetric equilibrium does not exist in the uniform auction for n2 ≥ 6 and in the discriminatory auction for n2 ≥ 7.
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
1618
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Figure A1
0.25
0.24
0.23
0.22
0.21
0.20
0.19
0.18
0.17
0.16
0.15
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
We now turn to the discriminatory auction and the
derivation of b d .
Consider −1/2 < x < 1/2. Then, by (A2),
Bidding Function in a Discriminatory Auction, b d x, for
Different Values of n2
n2 = 5
n2 = 6
n2 = 7
n2 = 8
vx x = b u x = x +
x
0
0.1
0.2
0.3
0.4
0.5
Note. Model parameters are specified in Example A.1.
Derivation of Example A.1
Note that f x v = 1 if x − v ≤ 1/2 and f x v = 0
if x − v > 1/2. Correspondingly, F x v = x − v + 1/2 if
x − v ≤ 1/2, F x v = 0 if x < v − 1/2, and F x v = 1 if
x > v + 1/2.
The support of the posterior distribution of v is bounded
by vx = max0 x − 1/2, vx = min
v x + 1/2. Equation (5)
becomes
vx M
1 ni −2
dv
i=1 i ni ni − 1 vx v x − v + 2
u
b x = vx M
1 ni −2
dv
i=1 i ni ni − 1 vx x − v + 2
Changing variables t = x − v + 1/2 yields dv = −dt,
v = x − t + 1/2, so
x−vx +1/2 M
1 ni −2
dt
i=1 i ni ni − 1 x−vx +1/2 x − t + 2 t
u
(A1)
b x =
x−vx +1/2
M
n
−2
i
dt
i=1 i ni ni − 1 x−vx +1/2 t
Consider x such that x < 1/2 < v − 1/2. In that case, x − vx +
1/2 = minx + 1/2 1 = x + 1/2, x − vx + 1/2 = 0, and (A1)
becomes
0 M
1 ni −2
dt
i=1 i ni ni −1 x+1/2 x −t + 2 t
u
b x =
0
M
n
−2
i
dt
i=1 i ni ni −1 x+1/2 t
n M
1
1
1 ni −1
− n1 x + 12 i
i=1 i ni ni −1 x + 2 ni −1 x + 2
i
=
M
1
1 ni −1
i=1 i ni ni −1 ni −1 x + 2
M
1 ni
1
i=1 i ni −1 x + 2
= x+ − (A2)
n −1 M
2
i ni x + 1 i
i=1
2
Consider x < 1/2. Using n1 = 2,
2
n2
1 1 x + 12 + 1 − 1 n2 − 1 x + 12
u
b x = x + −
n −1
2
1 2x + 1 + 1 − 1 n2 x + 1 2
= x+
1
2
2
2
1 1 x + + 1 − 1 n2 − 1 x + 12
−
n −2
2
21 + 1 − 1 n2 x + 12 2
n2 −1
As x 1/2, i.e., for x close enough to, but less than, 1/2,
this function is increasing for n2 ≤ 5 and is decreasing for
n2 ≥ 6. Therefore, by Theorem 2.1, an increasing symmetric
equilibrium exists for n2 ≤ 5, but no such equilibrium exists
if n2 ≥ 6. (This part of Example A.1 is also discussed in
Harstad et al. 2008.)
1
−
2
M
ni
i=1 i ni − 1x + 1/2
M
n
−1
i
i=1 i ni x + 1/2
By (6), using k = 1,
M
i=1 i ni fni −1 k y x
Ayx = M
i=1 i ni Fni −1 k y x
v
M
ni −2
y tgt dt
i=1 i ni v ni −1f x tf y tF
=
M
v
ni −1 y tgt dt
i=1 i ni v f x tF
For −1/2 < x < 1/2,
x+1/2
M
i ni ni − 1 0
x − t + 1/2ni −2 dt
Ax x = i=1M
x+1/2
x − t + 1/2ni −1 dt
i=1 i ni 0
M
ni −1
i=1 i ni x + 1/2
= M
n
i
i=1 i x + 1/2
Then,
d
b x =
=
x
−1/2
x
x
vttAttexp − Ass dt
−1/2
M
M
t
ni
i=1 i ni −1t +1/2
t +1/2− M
ni −1
i=1 i ni t +1/2
ni −1
i=1 i ni t +1/2
M
ni
i=1 i t +1/2
x M
ni −1
i=1 i ni s +1/2
ds dt
·exp −
M
ni
t
i=1 i s +1/2
M
M
x
ni
ni
i=1 i ni t +1/2
i=1 i ni −1t +1/2
−
=
M
M
ni
ni
−1/2
i=1 i t +1/2
i=1 i t +1/2
x M
ni −1
i=1 i ni s +1/2
ds dt
·exp −
M
n
i
t
i=1 i s +1/2
x M
x
ni −1
i=1 i ni s +1/2
exp −
ds
dt
=
M
ni
−1/2
t
i=1 i s +1/2
·
Substituting M = 2, n1 = 2, 1 = 2 = 1/2 yields bidding
functions depicted in Figure A1. As indicated by the figure, b d x is increasing in x with n2 = 5 and n2 = 6, and
decreasing for some x with n2 = 7 and n2 = 8. In general, it
can be shown that b d x is increasing in x when n2 ≤ 6 and
decreasing for some x when n2 ≥ 7. Example A.1 builds on the fact that vx x might not
be increasing. Note that, as shown in Milgrom and Weber
(1982), if vx x is increasing (e.g., in the private-value setting, where vx x = vni k x x = x), then b d x is increasing.
(As stated in Theorem 2.1, an increasing vx x is sufficient
to guarantee the existence of a symmetric equilibrium in
the uniform auction.) However, even if b d x given by (7) is
increasing, an increasing symmetric equilibrium still might
not exist in a discriminatory auction. The reason is that
deviations from (7) could be profitable:
Example A.2. Consider an auction for one object (i.e.,
k = 1). Let gv be uniform on 01 v, v > 1, let f x v be
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
1619
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
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uniform on v − 1/21 v + 1/2, and let uv x = x. Then,
for large enough v, an increasing symmetric equilibrium
2
in a discriminatory auction does not exist if M
i=1 i ni >
M
2
where n = i=1 i ni , which is equivalent to variance
2n + n,
of the number of bidders being greater than n2 + n.
Derivation of Example A.2
Denote y = b d −1 b. Then, using (6), the derivative of the
expected profit, given by Equation (B6), is
M
i=1 i ni fni −1 k y
d y
nb
x
M
i=1 i ni fni −1 k y
Ay x = M
x
i=1 i ni Fni −1 k y x
v
M
ni −2
y tgt dt
i=1 i ni v ni − 1f x tf y tF
=
v
M
n
−1
i
y tgt dt
i=1 i ni v f x tF
y+1/2
M
ni −2
dt
i=1 i ni ni − 1 x−1/2 y − t + 1/2
=
y+1/2
M
−1
n
i
dt
i=1 i ni x−1/2 y − t + 1/2
M
ni −1
i=1 i ni y − x + 1
(A3)
= M
n
i
i=1 i y − x + 1
Let 1/2 < y < v − 1/2. Because uv x = x, vx y = x. Then,
by (7),
y
1/2
tAttexp − Ass ds dt
b d y =
+
y
1/2
t
y
tAttexp − Ass ds dt
= exp −
t
y
1/2
·exp −
+
y
1/2
Ass ds
1/2
t
1/2
−1/2
tAtt
Ass ds dt
y
tAttexp − Ass ds dt
t
Note that, by (A3), As s = n for s > 1/2. Denoting
1/2
1/2
I=
tAt t exp −
As s ds dt
−1/2
b d y = exp −
y
1/2
n ds I +
+
= I exp1/2 − yn
y
1/2
y
1/2
t n exp −
y
t
n ds dt
dt
t n expt − yn
+ exp−y n
= I exp1/2 − yn
expy n
− 1/2 − 1/n
expn/2
· y − 1/n
= y − 1/n + exp−y n
vx y − b d y − b d y/Ay x
and vy y − b d y − b d y/Ay y = 0. Thus, a necessary
condition for the existence of an increasing symmetric equilibrium is that vx y − b d y − b d y/Ay x is nondecreasing in x when x is close enough to y. (Otherwise, a
bidder with signal y would increase her expected payoff by
bidding either slightly higher or slightly lower than b d y.)
Note that for the case of a known number of bidders this
is always the case: vx y is increasing in x and Ay x is
increasing in x by Lemma 1 of Milgrom and Weber (1982).
In the case of an unknown number of bidders, vx y is
still increasing in x, but Ay x might be decreasing in x.
Consider numerical Example A.2. We will show that, for
large enough y, vx y − b d y − b d y/Ay x might be
decreasing in x when x is close enough to y.
For 1/2 < y ≤ x < v − 1/2, using k = 1, we have
−1/2
we have
t
expn/2
· I expn/2
− 1/2 − 1/n
(A4)
Consider the limit v → and y → . Then, by (A4), b d y
so b d y → 1 as y → . Thus, using
approaches y − 1/n,
vx y = x and Ay x from (A3),
dvx y − b d y − b d y/Ay x/dx
M
y − x + 1ni
= d x − M i=1 i
dx
ni −1
i=1 i ni y − x + 1
M
i ni y − x + 1ni −1
= 1 + i=1
M
ni −1
i=1 i ni y − x + 1
M
M
ni
ni −2
i=1 i y − x + 1
i=1 i ni ni − 1y − x + 1
−
M
2
ni −1
i=1 i ni y − x + 1
For x = y, the above equation becomes
dvx y − b d y − b d y/Ay x/dx x=y
M
M
2
1
i ni ni − 1
i=1 i ni
= 2 − i=1
=
2
+
−
M
2
2
n
n
i=1 i ni
M
2
2 + n,
the quantity above is
Therefore, if
i=1 i ni > 2n
negative and a symmetric increasing equilibrium does not
exist. Appendix B. Proofs
Proof of Theorem 2.1. Harstad et al. (2008) prove the
theorem for the common-value case uv x = v. The proof
for the general form of uv x is analogous.
Assume that an increasing function b0 is a symmetric
equilibrium bid function. Hence, the bid b0 x maximizes
expected profit for a bidder who observes the signal x when
all rivals use b0 . Consider such a bidder observing signal x.
By assumption, b0 x is increasing but does not have to be
continuous. Define b0−1 b = supx" b0 x ≤ b, i.e., b0−1 b is
the largest x such that b0 x ≤ b. Then b0−1 is defined for all b
and, for all x, x = b0−1 b0 x. Conditional on the numbers of
bidders and objects ni ki , the expected profit 3i b x of a
bidder who observes x and bids b when all rivals use the
function b0 is
3i b x =
b0−1 b
x
vni ki x y − b0 yfni −1 ki y x dy
Using Matthews (1987), the probability of ni bidders, given
where n = M
that a bidder is active, is i ni /n,
i=1 i ni . The
ex ante unconditional expected profit for a bidder who
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
1620
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
observes signal x and bids b (i.e., taking the expectation
over all pairs ni ki ), is
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bx =
=
b0 x2 x1 − b0 x1 x1 x2
vx1 y − b0 yfY y x dy > 0
=
M
i ni
3i bx
n
i=1
−1
M
i ni b0 b
vni ki xy−b0 yfni −1ki y x dy
n x
i=1
x1
(B1)
Denote by fY the density of the pivotal rival signal, i.e.,
the ki th highest of the remaining ni − 1 signals where the
probability of ni ki pair is i ni /n:
fY y x =
M
i ni
y x
f
n ni −1 ki
i=1
Then (B1) can be expressed as
b x =
b0−1 b
x
vx y − b0 yfY y x dy
(B2)
where, obviously, fY y x is nonnegative.
Also note that vx y is increasing in its first argument,
and is continuous. To show this, rewrite (4):
v
vx y =
v
uv xf x vgy v dv
v
f x tgy t dt
v
xL < xU ⇒ vxL y = EuV xL xL ≤ EuV xL xU (B3)
so the claim follows.
To show that b u , given by (5), is a symmetric equilibrium if it is increasing, first note that setting x = y in (4)
yields b u = vx x (and b u is continuous). Suppose vx x is
increasing. Then, using (B2),
b x − b u x x
bu−1 b
vx y − vy yfY y x dy
=
x
so a bidder observing signal x1 benefits by deviating from
bidding b0 x1 to bidding b0 x2 . If b0 x0 is discontinuous
from the right at x0 , then there exists b1 , b0 x0 < b1 <
vx0 x0 . Note that by definition b0−1 b1 = b0−1 b0 x0 = x0 .
A bidder who observes signal x0 benefits by bidding b1
instead of b0 x0 in the case of a tie with the pivotal bid
(recall that ties are settled randomly): conditional on a pivotal bidder having signal x0 , the expected value is vx0 x0 and the price is bx0 < vx0 x0 .
Therefore, if b u given by (5) is increasing, it is the unique
symmetric equilibrium in increasing strategies. Proof of Theorem 2.2. Assume that bidder j, observing
signal xj , bids b d xj . Consider a bidder who observes signal x. The bidding function b d x is a symmetric Nash equilibrium if and only if the expected profit of the bidder who
observes signal x is maximized at b = b d x. Conditional on
the number of bidders ni , the expected profit 3i b of the
bidder who observes signal x and bids b is
3i b =
M
M
where gy v =
i=1 i ni fni −1 ki y vgv/
i=1 i ni ·
v
f
y tgt dt. Because f · v satisfies MLRP
v ni −1 ki
(Equation (1)), by Theorem 2.1 of Milgrom (1981), for
every nondegenerate prior distribution G with p.d.f. g and
every xL and xU in the support of X1 such that xL < xU ,
G· X = xU dominates G· X = xL in the sense of
first-order stochastic dominance. In particular, for Gy with
p.d.f. gy
≤ EuV xU xU = vxU y
x1 < x2 in the neighborhood of x0 such that for all y, x1 ≤
y ≤ x2 , b0 y < vx1 y and fY y x1 > 0. Then, using (B2),
x
vni k x y − bfni −1 k y x dy
i=1
Taking the expectation over all ni , the total expected profit
of the bidder who observes signal x and bids b is
3b =
M
i=1
=
M
i=1
i
ni
3 b
n i
i
ni b
n x
d −1 b
vni k xy−bfni −1k y xdy
(B5)
Differentiating (B5), we get
d3b
1
= d d −1
b b b
db
M
i=1
Thus, because vx y is increasing in its first argument, (B4)
is nonpositive for all b = b u x. Hence, the bid b u x maximizes expected profit for a bidder who observes the signal
x when all rivals use b u , i.e., b u is a symmetric equilibrium
bid function.
Finally, suppose there exists an increasing symmetric
equilibrium bid function b0 , b0 = b u , i.e., suppose that there
exists x0 such that b0 x0 = vx0 x0 . Consider the case
b0 x0 < vx0 x0 (the case b0 x0 > vx0 x0 is treated similarly). If b0 is continuous from the right at x0 , then there exist
b d −1 b
As shown by Matthews (1987), from the bidder’s point of
view, the probability of ni bidders given the observed signal
where
x is i ni /n,
M
n = i ni ·
(B4)
−
i
M
i=1
ni
v x b d −1 b − bfni −1 k b d −1 b x
n ni k
i
d −1
ni b b
fni −1 k y x dy
n x
(B6)
Substituting b = b d x into (B6), we get the following differential equation for b d x, where the condition
d3b/dbb=bd x = 0 is satisfied:
M
d
i=1 i ni vni k x x − b xfni −1 k x x
b d x =
x
M
i=1 i ni x fni −1 k y x dy
M
d
i=1 i ni vni k x x − b xfni −1 k x x
(B7)
=
M
i=1 i ni Fni −1 k x x
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
1621
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
Equation (B7) generalizes Equation (7) of Milgrom and
Weber (1982, p. 1107) to the case of multi-item auctions with
an unknown number of bidders.
Recalling definitions (4) and (6), Equation (B7) can be
rewritten as
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b d x = vx x − b d xAx x
As in Milgrom and Weber (1982), the boundary condition is
b d x = v x x
Then the solution of (B7) is
x
t
b d x =
vt tAt te x Ass ds dt
x
Proof of Proposition 2.4. By (6),
M
i=1 i ni fni −1 k x x
Ax x = M
i=1 i ni Fni −1 k x x
(B8)
Substituting f x v = hx − v, F x v = Hx − v, and diffuse prior gv into (2) yields
fn−1 k x x = -nk (B9)
+ x
n − 1!
Fn−1 k x x =
hy−vH n−k−1 y−v
k−1!n−k−1! − −
· 1−Hy−vk−1 hx−v dy dv
(B10)
Applying the change of variables Hy − v = t, and then
H x − v = z, to the right-hand side of (B10) we have
Fn−1 k x x =
n − 1!
k − 1!n − k − 1!
1 z
t n−k−1 1 − tk−1 dt dz
·
0
0
(B11)
Note that the right-hand side of (B11) is the probability that
a random variable z drawn from a uniform distribution on
0 1 is greater than t, which is the kth order statistic out of
n − 1 independent draws from the same uniform distribution on 0 1. By symmetry, this probability is k/n, i.e.,
k
Fn−1 k x x = n
Thus, by (B8), (B9), and (B12),
M
M
i=1 i ni -ni k
i=1 i ni -ni k
Ax x = M
=
k̄
i=1 i ni k/ni (B12)
=
x
−
M
i=1 i ni +ni k
t − % M
i=1 i ni -ni k
i=1 i ni -ni k t−x M
i=1 i ni -ni k /k̄ dt
e
for some z∗ , t1 ≤ H −1 z∗ ≤ t2 . Because hH −1 z∗ ≥ ' for
all z∗ , we get -nk ≥ ' for all n k.
j
Let Ym denote the jth order statistic from a set of m signals. In the case of a uniform auction, when the realized
, and
number of bidders is ni , the price per object is b u Ynk+1
i
thus, using (10), the auction revenue in a uniform auction is
M
M
M
k+1 j=1 j nj +nj k
u
u
k+1
(B14)
R = i kb Yni = i k Yni −% M
i=1
i=1
j=1 j nj -nj k
In the case of a discriminatory auction, when the realized
number of bidders is ni , k objects are sold at prices b d Yn1i ,
b d Yn2i b d Ynki . Using (12), the auction revenue in a discriminatory auction is
Rd =
k̄
Applying the identity (that holds for any B and any C > 0)
t=x
x
1 tC e t + BCetC dt = tetC + B −
C
−
t=−
1 Cx
e
= x+B−
C
M
with B = −% M
and C =
i=1 i ni +ni k /
i=1 i ni -ni k M
i=1 i ni -ni k /k̄ yields (12). M
i=1
M
i=1
(B13)
M
·
= hH −1 z∗ =
Substituting vt t from Proposition 2.3 and At t from
(B13) into (7), with x = −, yields
x
t
b d x =
vt tAt te x Ass ds dt
x
Remark. Levin and Smith (1991, Equation (3)) identify
a continuum of increasing symmetric equilibria in a diffuse prior setting. In the proof of Proposition 2.4 we consider the setting with a diffuse prior as a limit of the
distributions gv with the support bounded from below
(i.e., x > −), and we assume that the boundary condition
b d x = v x x holds. This approach yields linear bidding
strategies, as the bidding function in Example A.2 for large
enough y.
Proof of Theorem 3.1. Before comparing revenues, we
first show that -nk ≥ '. By (8), changing the integration variable to Ht
1 = z, and recalling that n − 1!/
k − 1!n − k − 1! 0 zn−k−1 1 − zk−1 dz = 1, we have
t2
n − 1!
-nk =
H n−k−1 t1 − Htk−1 h2 t dt
k − 1!n − k − 1! t1
1
n − 1!
hH −1 zzn−k−1 1 − zk−1 dz
=
k − 1!n − k − 1! 0
=
M
i=1
i
i
i
k
j=1
k
j
b d Yni j=1
l=1 l nl +nl k
l=1 l nl -nl k
j=1
k
M
j
Yni −% M
j
Yni −k
M
i=1
− M
k
l=1 l nl -nl k
i
M n +
k
j=1 j j nj k
· % M
+ M
j=1 j nj -nj k
j=1 j nj -nj k
(B15)
The difference in expected auction revenues, using (B14)
and (B15), is
ERu −ERd =
M
i=1
=
i
M−1
i=1
k
k2
j E Ynk+1
−Y
+
n
i
M
i
j=1
i=1 i ni -ni k
i
k
k
j j E Ynk+1
−Yni +M E Ynk+1
−YnM
i
M
j=1
+ M
k2
i=1 i ni -ni k
j=1
(B16)
M−1 k
k+1
−
Note that the first term in (B16),
i=1 i
j=1 EYni
j
Yni , is strictly negative and does not depend upon nM .
Pekeč and Tsetlin: Revenue Ranking of Auctions with an Unknown Number of Bidders
1622
Management Science 54(9), pp. 1610–1623, © 2008 INFORMS
j
The second term, M kj=1 EYnk+1
− Yn , is negative for
MM−1 Mk
j
∗
− Yni +
any nM . Let n be such that i=1 i j=1 EYnk+1
i
2
∗
∗
k /M n ' ≤ 0. Then, for any nM ≥ n ,
ERu − ERd <
M−1
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i=1
<
M−1
i=1
<
M−1
i=1
i
k
k2
j E Ynk+1
− Yni + M
i
j=1
i=1 i ni -ni k
i
k
j E Ynk+1
− Yni +
i
k2
M n∗ -nM k
i
k
j E Ynk+1
− Yni +
i
k2
≤ 0
M n∗ '
j=1
j=1
Proof of Theorem 4.2. If the number of bidders is
unknown, bidders bid according to (10) and (12) in uniform
and discriminatory auctions, respectively. However, if the
number of bidders is revealed before the bids are submitted, the bidding strategies are given by (9) and (11). In the
for ni biduniform auction, the price per object is bndi k Ynk+1
i
ders and k objects, and thus the uniform auction revenue
when the number of bidders becomes known prior to bid
submission is
M
M
+ni k
k+1
−
%
Y
(B17)
=
k
RuK = k i bnui k Ynk+1
i
ni
i
-ni k
i=1
i=1
In a discriminatory auction, in the case of ni bidders,
k objects are sold at prices bndi k Yn1i bndi k Yn2i bndi k Ynki .
Using (11), the auction revenue in a discriminatory auction is
k
k M
M
+n k
k
j
j
i bndi k Yni = i
Yni − % i −
RdK =
-ni k ni -ni k
i=1
j=1
i=1
j=1
=
M
i=1
i
k
j=1
j
Yni − k
+n k
k
i % i +
-ni k ni -ni k
i=1
M
(B18)
To study the effect of revealing the number of bidders on
auction revenues, we compare ERu with ERuK and ERd with ERdK .
From (B14) and (B17),
M
M
+n k
i=1 i ni +ni k
− k i i ERuK − ERu = % k M
-ni k
i=1
i=1 i ni -ni k
From (B15) and (B18),
ERdK − ERd M
+n k
% M
k
i=1 i ni +ni k + k
− k i % i +
= k M
-ni k ni -ni k
i=1
i=1 i ni -ni k
M
M
k M
k
j=1 j
= ERuK − ERu + k i M
− k i
n
n
i ni k
i=1
i=1
j=1 j j nj k
2
M
1
1
= ERuK − ERu − k2 i ni -ni k
− M
ni -ni k
i=1
j=1 j nj -nj k
≤ ERuK − ERu References
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